Fix for mrDMD division by zero error during reconstruction about pydmd HOT 2 CLOSED

Dear @rfmiotto, thank you for the effort! We'll check the Merge Request as soon as possible.

Regarding your dubts:

• the dictionaries initialization is performed for all the DMD classes, only to initialize the attributes (nothing related with the MrDMD prediction).
• if I remember correctly, the try:...except lines have been added in order to avoid errors that occur when the step variable is too large to fit in a single window. Better (and cleaner) solutions are very welcome!

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Dear @ndem0,

Thank you for clarifying.

But can I say that the mrDMD cannot handle predictions? (the dmd_time is a copy of original_time)

The solution that I propose in this PR is to limit the maximum level such that the bins at the topmost level are composed of at least 4 snapshots.

# Redefine max level if it is too big.
lvl_threshold = int(np.log(self._snapshots.shape[1]/4.)/np.log(2.))
if self.max_level > lvl_threshold:
    self.max_level = lvl_threshold
    print('Too many levels. Redefining `max_level` to {}'.format(self.max_level))

Also, I ensured that the minimum admissible step was 1, since slice step cannot be zero:
step = max(1, int(np.floor(old_div(n_samples, nyq))))

The reason for doing that is because (and please, correct me if I am wrong):

In the original code (I mean before this proposed commit), as we go to higher resolution levels the n_samples decreases up to a point when the n_samples/nyq becomes < 1. At this point, the step variable becomes zero, causing a ValueError to be raised in Xsub = Xraw[:, ::step]. So the code goes to the exception block and sets the DMD variables (modes, eigenvalues, etc) of that bin as zero.

This approach solves for the zero slice step in Xsub = Xraw[:, ::step], but when one tries to reconstruct the dynamics, these zero-valued bins are included in the computation and cause a division by zero error.

To my understanding, the try:..except blocks were added to solve this step problem. However, I didn't understand what exactly did you mean by the try:...except to be related to the step being large to fit a window. For example, the code below gives no errors:

a = np.arange(10)

print( a[::7] )  # Out: array([0, 7])
print( a[::25] ) # Out: array([0])

Please, let me know if you agree with that or if my conclusions are wrong.

Thank you and best regards,
Renato Miotto

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